Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies
This addresses a problem for researchers and practitioners in statistics and machine learning who need to model real-world datasets with positive dependencies, though it is incremental as it builds on existing graphical model frameworks.
The paper tackles the limitation of previous multivariate graphical models that could not model positive dependencies for exponential and Poisson distributions, and introduces Square Root Graphical Models (SQR) that allow for almost arbitrary positive and negative dependencies with mild constraints, demonstrating this on synthetic and real-world airport delay data.
We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. However, in many real-world datasets, variables clearly have positive dependencies. For example, the airport delay time in New York---modeled as an exponential distribution---is positively related to the delay time in Boston. With this motivation, we give an example of our model class derived from the univariate exponential distribution that allows for almost arbitrary positive and negative dependencies with only a mild condition on the parameter matrix---a condition akin to the positive definiteness of the Gaussian covariance matrix. Our Poisson generalization allows for both positive and negative dependencies without any constraints on the parameter values. We also develop parameter estimation methods using node-wise regressions with $\ell_1$ regularization and likelihood approximation methods using sampling. Finally, we demonstrate our exponential generalization on a synthetic dataset and a real-world dataset of airport delay times.